Thread EQing vocals...

For one thing, make sure you're not EQing them without the context of the rest of the tracks. As the author of our Mixing series pointed out in the the most recent installment (https://en.audiofanzine.com/recording-mixing/editorial/articles/the-eqing-primer.html), EQ decisions made only from listening in solo are usually not as good as those made with the other tracks also playing. Other than that, I don't see anything wrong with your method, as long as its working for you. Eventually, you'll get to the point where you can pretty closely distinguish the particular frequencies are that you're hearing, and you'll be able to zero in on the unpleasant ones without having to go through as much listening as you're doing now. But it's certainly not unusual to boost at a particular frequency and then "sweep" it across the frequency band to zero in on a particular troublesome area.

Another thing you have to be very careful about with vocals, especially if you're boosting frequencies on them, is that you don't accentuate sibilance, which mainly occurs in the 4-10 kHz range. Of course, if your track is overly sibilant to start with, try using a de-esser on it.

One other point about vocal EQ: I find it helpful to use a high-pass filter (which allows audio to pass above a user specified frequency) to roll off the frequencies below the voice, to reduce the potential muddiness. I typically set the filter low, and then slowly bring it up while listening to its affect on the track, and when I get to the point where I'm cutting too much and the voice starts to thin out, I'll back it off. This is also useful to do on your other tracks. Read more about this subject here: https://en.audiofanzine.com/recording-mixing/editorial/articles/cut-the-mud-from-your-mix.html

I always seem to get a very unpleasant, excessive boominess between 250-380Hz in my vocals. I always wondered whether that's normal in vocals, whether it's a problem in my mic setup, whether it's a problem with my mic, or whether it's an issue unique to my vocals...

There's a lot of variables that could be at play. Mic type, mic placement and your microphone technique, and room acoustics. Have you recorded your voice in other studios? Does the same thing happen?